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arXiv:1912.03071 · International Journal of Geometric Methods in Modern Physics

Isotropic Lifshitz scaling in four dimensions

Dario Zappalà

Published Feb 11, 2020• 6 citations
arXivHTMLPDF

Abstract

The presence of isotropic Lifshitz points for a [Formula: see text]-symmetric scalar theory is investigated with the help of the Functional Renormalization Group. In particular, at the supposed lower critical dimension [Formula: see text], evidence for a continuous line of fixed points is found for the [Formula: see text] theory, and the observed structure presents clear similarities with the properties observed in the two-dimensional Berezinskii–Kosterlitz–Thouless phase.

DOI
10.1142/s021988782050053x

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The presence of isotropic Lifshitz points for a [Formula: see text]-symmetric scalar theory is investigated with the help of the Functional Renormalization Group. In particular, at the supposed lower critical dimension [Formula: see text], evidence for a continuous line of fixed points is found for the [Formula: see text] theory, and the observed structure presents clear similarities with the properties observed in the two-dimensional Berezinskii–Kosterlitz–Thouless phase.
The presence of isotropic Lifshitz points for a U(1)U(1)U(1) symmetric scalar theory is investigated with the help of the functional renormalization group at the conjectured lower critical dimension d=4d=4d=4. To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the renormalization group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found that present evident similarities with the properties shown by the line of fixed points observed in the two-dimensional Berezinskii-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at a large distance of the order parameter correlations.
Abstract The space of solutions of the exact renormalization group fixed point equations of the two-dimensional RP N −1 model, which we recently obtained within the scale invariant scattering framework, is explored for continuous values of N ⩾ 0. Quasi-long-range order occurs only for N = 2, and allows for several lines of fixed points meeting at the Berezinskii–Kosterlitz–Thouless transition point. A rich pattern of fixed points is present below N * = 2.244 21‥, while only zero temperature criticality in the O ( N ( N + 1)/2 − 1) universality class can occur above this value. The interpretation of an extra solution at N = 3 requires the identification of a path to criticality specific to this value of N .
A new renormalization group treatment is proposed for the critical exponents of an m-fold Lifshitz point. The anisotropic cases (m not equal 8) are described by two independent fixed points associated to two independent momentum flow along the quadratic and quartic directions, respectively. The isotropic case is described separately. In that case, the fixed point is due to renormalization group transformations along the quartic directions. The new scaling laws are derived for both cases and generalize the ones previously reported.
We employ the derivative expansion of the nonperturbative renormalization group to address the phenomenon of anisotropic scale invariance and the associated functional fixed points, also known as Lifshitz points, in systems characterized by a scalar order parameter. We demonstrate the existence of the Lifshitz fixed point featuring a nonclassical value of the anisotropy exponent θ<1/2 and provide estimates for values of a set of critical exponents in the physically most relevant case of the three-dimensional uniaxial Lifshitz point (d,m)=(3,1), m denoting the anisotropy index. We compare our predictions with existing estimates from perturbative expansions around dimensionality d=4+1/2 as well as those from the 1/N expansion.
Field-theoretic renormalization-group methods are developed to describe in a unified fashion the critical exponents of an m-fold Lifshitz point at the two-loop order in the anisotropic $(m\ensuremath{\ne}d)$ and isotropic (m=d(m=d(m=d close to 8) situations. The general theory is illustrated for the N-vector ${\ensuremath{\varphi}}^{4}$ model describing a d-dimensional system. A new regularization and renormalization procedure is presented for both types of Lifshitz behavior. The anisotropic cases are formulated with two independent renormalization-group transformations. The description of the isotropic behavior requires only one type of renormalization-group transformation. We point out the conceptual advantages implicit in this picture and show how this framework is related to other previous renormalization-group treatments for the Lifshitz problem. The Feynman diagrams of arbitrary loop order can be performed analytically provided these integrals are considered to be homogeneous functions of the external momenta scales. The anisotropic universality class (N,d,m)(N,d,m)(N,d,m) reduces easily to the Ising-like (N,d)(N,d)(N,d) when m=0.m=0.m=0. We show that the isotropic universality class (N,m)(N,m)(N,m) when m is close to 8 cannot be obtained from the anisotropic one in the limit $\stackrel{\ensuremath{\rightarrow}}{d}m$ near 8. The exponents for the uniaxial case d=3,N=m=1d=3,N=m=1d=3,N=m=1 are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.